Using the ROI data set:
For each of the two majors:
Draw the scatter diagram of Y = ‘Annual % ROI’ against X = ‘Cost’.
The obtained two diagrams are given below,
Obtain b0 and b1 of the regression equation defined as y ̂ = b0 + b1X and the coefficient of determination (r2) from the Excel regression output.
For Business Major, the obtained output is given below.
| SUMMARY OUTPUT | ||||||
|---|---|---|---|---|---|---|
| Multiple R | 0.9701 | |||||
| R Square | 0.9410 | |||||
| Adjusted R Square | 0.9377 | |||||
| Standard Error | 0.0027 | |||||
| Observations | 20 | |||||
| ANOVA | ||||||
| df | SS | MS | F | Significance F | ||
| Regression | 1 | 0.0022 | 0.0022 | 287.2207 | 0.0000 | |
| Residual | 18 | 0.0001 | 0.0000 | |||
| Total | 19 | 0.0023 | ||||
| Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | |
| Intercept | 0.1180 | 0.0024 | 48.5862 | 0.0000 | 0.1129 | 0.1231 |
| Cost | -2.11204E-07 | 1.24622E-08 | -16.94758619 | 1.64456E-12 | -2.37386E-07 | -1.85022E-07 |
From the above output we can see that b0 = 0.1180 and b1= -2.11204E-07 thus the regression equation is,
Y = 0.1180-2.11204E-07*X
For Engineering Major, the obtained output is given below.
| SUMMARY OUTPUT | ||||||
|---|---|---|---|---|---|---|
| Multiple R | 0.9754 | |||||
| R Square | 0.9515 | |||||
| Adjusted R Square | 0.9488 | |||||
| Standard Error | 0.0033 | |||||
| Observations | 20 | |||||
| ANOVA | ||||||
| df | SS | MS | F | Significance F | ||
| Regression | 1 | 0.0039 | 0.0039 | 352.8738 | 0.0000 | |
| Residual | 18 | 0.0002 | 0.0000 | |||
| Total | 19 | 0.0041 | ||||
| Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | |
| Intercept | 0.1268 | 0.0020 | 62.7372 | 0.0000 | 0.1225 | 0.1310 |
| Cost | -2.1455E-07 | 1.14214E-08 | -18.78493483 | 2.83396E-13 | -2.38545E-07 | -1.90554E-07 |
From the above output we can see that b0 = 0.1268 and b1= -2.1455E-07 thus the regression equation is,
Y = 0.1268-2.1455E-07*X
Draw the fitted regression line on the scatter diagram.
The plots are given below,
Calculate the estimated ‘Annual % ROI’ when the ‘Cost’ (X) is $160,000.
The estimated ‘Annual % ROI’ when the ‘Cost’ (X) is $160,000 is,
For Business Major = 8.42% and for Engineering Major = 9.25%
Test the hypothesis:
H0: β1 = 0
Ha: β1 ≠ 0
From the above output, the p-values for both the Business Major and Engineering Major are very small indicating that the above null hypothesis is rejected.
Write a paragraph or more on any observations you make about the regression estimates, coefficient of determination, the plots, and the results of your hypothesis tests.
In working this out, both the regression model are significant at 5% significance level (the F test has small p-value). The coefficient of determination indicated that the models are good fit to the data. Thus the above models can be used to predict the dependent variable Annual ROI from the independent variable cost.